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Theorem tsbi2 35565
 Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsbi2 (𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))

Proof of Theorem tsbi2
StepHypRef Expression
1 pm5.21 823 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
21olcd 871 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓)))
3 pm4.57 988 . . . . 5 (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑𝜓))
43biimpi 219 . . . 4 (¬ (¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
54orcd 870 . . 3 (¬ (¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓)))
62, 5pm2.61i 185 . 2 ((𝜑𝜓) ∨ (𝜑𝜓))
76a1i 11 1 (𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845 This theorem is referenced by:  tsxo2  35569  mpobi123f  35593  mptbi12f  35597  ac6s6  35603
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